Asymptotic power series

We consider the asymptotic behaviour of the second discrete Painlevé equation in the limit as the independent variable becomes large. Using asymptotic power series, we find solutions that are asymptotically pole-free within some region of the complex plane. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. We subsequently apply exponential asymptotic techniques to investigate such phenomena, and obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the behaviour of these asymptotic solutions shares a number of features with the tronquée and tri-tronquée solutions of the second continuous Painlevé equation.

We consider the asymptotic behavior of solutions of the first -difference Painlevé equation in the limits and . Using asymptotic power series, we describe four families of solutions that contain free parameters hidden beyond-all-orders. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. In order to investigate such phenomena we apply exponential asymptotic techniques to obtain mathematical descriptions of the rapid switching behavior associated with Stokes curves. Through this analysis, we also determine the regions of the complex plane in which the asymptotic behavior is described by a power series expression, and find that the Stokes curves are described by curves known as -spirals.

This technical paper of mathematical physics arose as an aftermath of the 2002 Cassini experiment (Bertotti et al 2003 Nature 425 374–6), in which the PPN parameter γ was measured with an accuracy σγ = 2.3 × 10−5 and found consistent with the prediction γ = 1 of general relativity. The Orbit Determination Program (ODP) of NASA's Jet Propulsion Laboratory, which was used in the data analysis, is based on an expression (8) for the gravitational delay Δt that differs from the standard formula (2); this difference is of second order in powers of m—the gravitational radius of the Sun—but in Cassini's case it was much larger than the expected order of magnitude m2/b, where b is the distance of the closest approach of the ray. Since the ODP does not take into account any other second-order terms, it is necessary, also in view of future more accurate experiments, to revisit the whole problem, to systematically evaluate higher order corrections and to determine which terms, and why, are larger than the expected value. We note that light propagation in a static spacetime is equivalent to a problem in ordinary geometrical optics; Fermat's action functional at its minimum is just the light-time between the two end points A and B. A new and powerful formulation is thus obtained. This method is closely connected with the much more general approach of Le Poncin-Lafitte et al (2004 Class. Quantum Grav. 21 4463–83), which is based on Synge's world function. Asymptotic power series are necessary to provide a safe and automatic way of selecting which terms to keep at each order. Higher order approximations to the required quantities, in particular the delay and the deflection, are easily obtained. We also show that in a close superior conjunction, when b is much smaller than the distances of A and B from the Sun, say of order R, the second-order correction has an enhanced part of order m2R/b2, which corresponds just to the second-order terms introduced in the ODP. Gravitational deflection of the image of a far away source when observed from a finite distance from the mass is obtained up to O(m2).

A uniformly valid asymptotic expansion is obtained for the solution of the one-dimensional time-independent Schrödinger equation which describes an anharmonic oscillator. The computational method is suggested by McKelvey's treatment of the two-turning-point problem, but has some practical advantages over his procedure. For turning points which lie close together, the energy levels are expressed as an asymptotic power series in ℏ/ (2m)1/2, with coefficients which depend only on the quantum number n and the derivatives of the potential at its minimum. The leading terms of this series are calculated and found to agree with the JWKB and perturbation results of earlier authors. For turning points which do not lie close together, the application of this method is indicated briefly and pursued up to the recovery of the Bohr—Sommerfeld quantum condition.

Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = - \sum_{\nu=0}^{\infty} a_{\nu}$ of infinite series $\sum_{\nu=0}^{\infty} a_{\nu}$ for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation $\Delta r_n = a_{n+1}$. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding Euler-Maclaurin formula. In the case of the other series considered -- the Gaussian hypergeometric series ${}_2 F_1 (a, b; c; z)$ and the divergent asymptotic inverse power series for the exponential integral $E_1 (z)$ -- the corresponding linear equations are solved symbolically with the help of Maple. The practical usefulness of the new formalism is demonstrated by some numerical examples.

This article is concerned with the asymptotic behaviour of m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation y“ + (λ − q(x))y = 0 on [0, ∞), as λ →∞ in a sector in the upper half of the complex plane. It is assumed that the potential function q is integrable near 0. A simplified proof is given of a result of Atkinson [7], who derived the first two terms in the asymptotic expansion of m(λ), and a sharper error bound is obtained. Theproof is then generalised to derive subsequent terms in the asymptotic expansion. It is shown that the Titchmarsh-Weyl m-coefficient admits an asymptotic power series expansion if the potential function satisfies some smoothness condition. A simple method to compute the expansion coefficients is presented. The results for the first few coefficients agree with those given by Harris [9].

From the point of view of physics, there are essentially two problems in the study of quantum field models. The first problem is to find the phenomena of interest to physics (e.g. particles, bound states, resonances, broken symmetries, and asymptotic power series expansions) in the two and three dimensional quantum field models.1 This problem includes obtaining the mathematical structure underlying these phenomena, and determining qualitatively which interactions and/or parameter values give rise to these phenomena and which do not. We believe this problem is feasible at the present time; indeed the initial progress is the subject of the present lectures. The second problem is the construction of nontrivial fields in four space time dimensions. Here it seems that some new ideas are needed to supplement those which have already been used in the construction of quantum fields.

<p style='text-indent:20px;'>We develop a Korteweg–De Vries (KdV) theory for weakly nonlinear waves in discontinuously stratified two-layer fluids with a generally prescribed rotational steady current. With the help of a classical asymptotic power series approach, these models are directly derived from the divergence-free incompressible Euler equations for unidirectional free surface and internal waves over a flat bed. Moreover, we derive a Burns condition for the determination of wave propagation speeds. Several examples of currents are given; explicit calculations of the corresponding propagation speeds and KdV coefficients are provided as well.</p>

A spatial and angular moments analysis of the monoenergetic pencil beam problem

The authors present a numerical analysis of the Stark-Wannier states in finite Kronig-Penney crystals extended to the complex field plane. Exponential stability in the infinite crystal limit for any fixed complex field value f is verified. The energies En.j(f) of such states are all connected through sequences of couples of branch points of the Bender-Wu type approaching symmetrically the real axis. On the other hand, near f=0 in the complex directions they have a smooth behaviour and it is possible to compute the first terms of the asymptotic power series.

For a class of nonlinear oscillation problems described by a second order ordinary differential equation containing a small nonnegative perturbation parameter $\varepsilon $ we determine asymptotic solutions which are uniformly valid as $\varepsilon \downarrow 0$ on intervals of the order of magnitude $\varepsilon ^{ - 1} $. These asymptotic solutions are obtained from formal asymptotic solutions subject to a condition in order that they approximate the exact solution asymptotically. These formal asymptotic solutions are constructed in the form of a finite generalized asymptotic power series involving functions of the periodic two variable type.

In the paper, the authors establish explicit formulas for asymptotic and power series expansions of the exponential and the logarithm of asymptotic and power series expansions. The explicit formulas for the power series expansions of the exponential and the logarithm of a power series expansion are applied to find explicit formulas for the Bell numbers and logarithmic polynomials in combinatorics and number theory.

The second part of the article discusses the accuracy of generalized asymptotic series representing the stress and displacement fields associated with the tip of an acute crack in orthotropic materials in the formulation of the plane problem of the theory of anisotropic elasticity. We compare the exact analytical solution to the problem of stretching an infinite anisotropic plane with an inclined central crack obtained using methods of the complex variable theory and an approximate solution found using the power series expansion method. For the first time, the fields of absolute error allowed for the truncation of the asymptotic series on a different number of terms for materials with a cubic symmetry of elastic properties are obtained. The analysis of the fields of the decadic logarithm of absolute errors showed that near the crack tip of all types of combined (mixed) deformation there are geometric points – loci of accuracy, in which the approximate solution almost coincides with the exact solution, which can be used in the interpretation of experimental and computational data obtained in order to determine the coefficients of asymptotic series for stresses and displacements near the tip of a crack or notch. To quantify the errors allowed when truncating the asymptotic series on the th term, a relatively accurate analytical solution obtained on the basis of the theory of the function of a complex variable the -norm is introduced into scrutiny, which gives an opportunity to choose and specify the number of terms of the series necessary to achieve the required accuracy when presenting the asymptotic ansatz with a multi-coefficient asymptotic series, for a wide range of slope angle values of the crack to the vertical axis (the axis of action of the applied tensile load) and the angle, setting the location of the anisotropy axes of the elastic properties of the material.

Helicon wave (whistler waves bounded transversely by a magnetic field surface) propagation is investigated for a cylindrically symmetric curved (flaring) boundary using a finite-element method where the propagation region is divided into a sequence of truncated cones. In each conical segment, a local spherical coordinate system is used with the origin at the apex of the cone. A vector wave equation for the fields is formulated for a cold plasma and reduced, in spherical coordinates, to a pair of coupled partial differential equations (PDEs) for two scalar functions. The PDEs are separable for the azimuthal eigenvalue m = 0. The -dependence is a Legendre function, with non-integer eigenvalues determined by the cone angle. The dependence on the radial coordinate, x, satisfies a fourth-order ordinary differential equation (ODE). A straightforward numerical integration of the equation from to fails for large values of , because of the existence of an exponentially growing solution to the equation. Consequently, a different approach is needed. Four independent solutions, valid for , in the form of power series in x (PS) are obtained which are each asymptotically proportional to as . For x large, four asymptotic expansions (AE) are obtained in the form of series in 1/x times or . The problem then is to find four linear combinations of the four PSs, each of which approaches one of the AEs in the limit by first deriving a double-integral representation of each PS, valid for all x, and then using its limit to match each AE to a linear combination of the PSs. For special values of or 4N, N an integer) closed-form solutions (polynomials times exponentials) result that are exact representations of the solution for all x. Solutions are computed and illustrated for an outgoing wave using the closed-form solution for special values of , and as a function of and x using the PS and AE, which have a large range of overlap. The propagating solutions of the ODE are also obtained using a WKB method and are used to calculate the propagation in a slowly diverging parabola of revolution.

An explicit numerical-analytical method is demonstrated for accurate solving the Schrödinger equation in those cases when this equation reducible to a system of n coupled ordinary differential equations with singular points. Fundamental system of solutions is constructed as algebraic combinations of power series, power functions and logarithmic function in the neighbourhood of the regular singular point and as asymptotic expansions of solutions in the neighbourhood of the irregular singular point. The method is based on the calculation of recurrent sequences of constant matrices of coefficients in power series and in inverse power series in asymptotic expansions using derived recurrent relations, that makes possible to calculate solutions at any given point using only algebraic computations and elementary functions. In turn it makes possible to solve accurately the eigenvalue problem and scattering problem and to derive analytical expressions for the wavefunctions. The method is applied to calculations of energies and wavefunctions of the discrete spectrum and wave functions of the continuous spectrum of the hydrogen-like atoms and of acceptors in semiconductors.